Maximal commutative subalgebras A of M_n(K) satisfy dim A ≥ n for n ≤ 13, Courter's n=14 example is optimal, and stack constructions attain the bound for all n ≥ 14.
On the minimal dimension of maximal commutative subalgebras of $M_6(k)$
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abstract
We study the minimal dimension of maximal commutative subalgebras of the matrix algebra $M_n(k)$ over an algebraically closed field. While examples with dimension strictly smaller than n are known for $n \geq 14$, no such examples are known in smaller dimensions. In this paper, we show that for n = 6 every maximal commutative subalgebra $A\subset M_6(k)$ satisfies $\dim A \geq 6$. The proof is based on a detailed analysis of local algebras and their module structure, combined with explicit estimates of the dimension of the centralizer.
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Minimal Dimensions of Maximal Commutative Matrix Algebras and Sharp Courter-Type Bounds
Maximal commutative subalgebras A of M_n(K) satisfy dim A ≥ n for n ≤ 13, Courter's n=14 example is optimal, and stack constructions attain the bound for all n ≥ 14.